Ep 39: Everybody Learns, Everybody Grows

Pam Harris  00:01

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Kim Montague  00:08

And I'm Kim. 

Pam Harris  00:09

And we're here to suggest that mathematizing is not about mimicking, or rote memorizing. But it's about thinking and reasoning; about creating and using mental, mathematical relationships. That math class can be less like it has been for so many of us and more like mathematicians working together. We answer the question, if not algorithms, then what?

Kim Montague  00:35

So Pam, when we started this podcast, we knew that we wanted to talk right away about some of the things that we believe in, right? We wanted listeners to get a feel for what we were going to be sharing. So they even knew if the podcast is going to be for them. 

Pam Harris  00:50

Right, totally. 

Kim Montague  00:51

And in some of the first episodes, we tried to lay out some of the underlying principles: that Math is Figure-Out-Able, all about the Development of Mathematical Reasoning - that was really important. That it is about developing reasoning, but not just answer getting. 

Pam Harris  01:07

Very important.

Kim Montague  01:08

And then in planning recently, we realized that we haven't talked about something that was super important.

Pam Harris  01:13

I know, right? Like one of our bedrock parts of what we believe and informs everything we do. Alright, so today, y'all, we're going to talk about something that on the team, we affectionately call the finger thing, because I use my fingers to sort of demonstrate what I'm talking about. Since you can't see my fingers, I'm gonna try to describe kind of what it looks like and what we mean. But it's all about the idea that we can teach so that everybody learns, and everybody grows all the time with everything we do. 

Kim Montague  01:47

Yeah

Pam Harris  01:47

So today's episode we are calling, Everybody Learns, Everybody Grows. 

Kim Montague  01:52

And I think we can acknowledge that we all want that, right? We all want everyone to learn and grow in our class. That's not all that profound. But at the point we start getting into the world of differentiation, and for some people, it looks like maybe three different tasks in your room. Or maybe it looks like creating extra projects and games and puzzles for the fast finishers, while maybe cutting out some of the problems or part of the assignment for the kids who just need more time. And I'm not knocking those ways to differentiate, but you mean something different entirely. 

Pam Harris  02:24

Totally.

Kim Montague  02:24

Okay, so this is kind of your stick. So I'm just gonna ask you a few questions. Okay?

Pam Harris  02:29

All right. 

Kim Montague  02:29

But first, what is it not? 

Pam Harris  02:31

Alright, so let me describe what we see everywhere, kind of what we experienced. It's a way of thinking about a typical year of schooling. So if you sort of think about your typical year, you're a teacher, you're about to begin to think about that year. If you think about how you were taught, then it felt a lot like this to me, that the teacher sort of said, "Okay, my students are here. And I'm going to teach whatever that first skill is." Like, I think about my year, and I think about all the standards, I think about what they need to know. And I'm like, "Okay, here's the first thing they need to know. And so I'm going to teach it." So I te-e-each that to mastery. 

Kim Montague  03:09

Step one.

Pam Harris  03:10

Step one, teach it to mastery. We do this thing, I make sure they can all do it. I do, we do, you do, they can all do it. And then I say to myself, "Okay, now that they've all mastered it, now it's time to move on to the next skill." So whatever that next skill is, I might turn the page, I might look at a list, I might be like, whatever it is. I'm like, "Okay, now I'm going to teach the next skill." And I te-e-each that next skill to mastery. And the reason I go "te-e-each that next skill" is because it depends. Like I might spend a little bit of time on it. I might have to spend a couple days on it. I might have to do some mo-o-ore work with it until they've all kind of like got it. They're all getting correct answers. And it's kind of this linear trajectory, and I don't move on until they got it and then I move on. And I do the next skill and then I teach it to mastery to move on. And it's this linear trajectory, because it's skill, then skill, then skill. And I can think about it kind of linearly, like, in a row, in order. Once I've done this skill, then I do the next thing. And many of you are like, "Yeah, that's what we do, right? We sort of have this idea of the the kind of order that things go in, and we don't move on until something's mastered." 

Kim Montague  04:12

Or maybe that's what we did when we were in school even. Right?

Pam Harris  04:16

Exactly, yeah, that's kind of what we experienced. And so when I kind of talk about traditional education in a huge way, I'm kind of talking about the way that I was taught. So from my perspective, that's what it felt like to be a student that the teacher said, "Alright, here's the next thing and we teach it to mastery. Now you've all got it, I'm gonna move on to the next thing." It was this like step by step, linear trajectory, just kind of trajectory of what we do throughout the year.

Kim Montague  04:40

So let's talk about the outcomes of that.

Pam Harris  04:43

So if I say to myself, "Okay, I'm going to teach this to mastery." I kind of like wait until everybody's sort of got it, right? Like that might look like we did it today, and I give it to you for homework, and then tomorrow we go over some things and maybe I'll give you a quiz and then we kind of move on. Everybody's got it, we move on. But we actually have to kind of admit that in reality, not everybody had it to mastery.

Kim Montague  05:08

 Sure.

Pam Harris  05:08

You know, we do our best, right? We pull small groups, we send kids to intervention, but at some point, we have to move on to something new, and just kind of hope that they get it next year. Or, you know, in their pullout group or something. There are some other outcomes to that. So one outcome is, some students fall behind. They just can't kind of hang on to what's happening. Maybe they look like they've got it Tuesday, but by Wednesday, they're not good memorizers, they don't kind of hang on to it. So some students don't hang on. But there are other outcomes. Also, that some students are bored. Some students have already done the thing, they got it really quickly and they're sort of bored in class. So as we do kind of this linear trajectory - next skill, mimic, mimic, memorize next skill, mimic, step by step memorize as we do that kind of linear thing - some students fall behind, and some students are bored. If you saw me in person, the reason we call this the finger thing is that I usually sort of have on my one hand my index finger kind of pointing sideways to my other hand. And so this is where all the kids are. And since they're here, then I'm going to do the first skill of the year. And then I put that first skill of the year is my other hand, kind of pointing at that finger, but it's a little bit higher, because it's the skill, and they don't know it yet. And so I teach that to mastery, that skill, it's a little higher, and now they all own it. Now I raised my finger that represents all the kids to that first sort of level, they've got it, they're there. And then now we're excellent, we're there. So then it worked, it's time for the next skill. And so that's kind of above again, I put my finger above it, and I teach to mastery, and then all the kids kind of match that and so they're sort of lining up. Kids in this one finger, and they're sort of lined up, and they're all kind of doing the same thing. And it's this linear trajectory. And it's all based on mastery, and mimicry and having kids like mimic and memorize until they're there. And then the finger kind of moves up and they'll sort of line up. So they've got it and I move on to the next skill on my left finger moves up and we sort of go on from there.

Kim Montague  07:11

It's kind of like a stepladder. Right? 

Pam Harris  07:13

Yeah. 

Kim Montague  07:14

So that sounds fabulous. If all kids were exactly the same, right?

Pam Harris  07:19

Weren't yours, Kim?

Kim Montague  07:20

Yeah they all learned exactly one skill at a time.

Pam Harris  07:22

Yeah, cuz your kids were all exactly in the same place, right? So this is when if you could picture my fingers, I'm sort of pointing my two index fingers at each other. And I say to everybody in the world, there's a problem with this, because your kids aren't here. And I sort of emphasize this one finger, the kids aren't here. And then I kind of splay out my whole hand because your kids are here. They're everywhere. They're all over the place. They have different mathematical backgrounds, they're not the same students, will never be the same. They have different experiences, they have different interests. They also have different ways of seeing the world. Some kids are more verbal, some prefer writing or artistic expression, like my daughter, students are all over the place in so many ways. So it's not about like this one finger can describe all my students. Like I wish I had more than all my fingers that I could put up that like students are sort of all over the place. And because of that, it's not this kind of, "Ooh, here, my kids. I'm going to do this one thing and this one thing is going to work for all of them." Kids in reality, and you know this, you guys are human beings, you're teachers, you know that your students all over the place.

Kim Montague  08:31

Yeah. So here's the question. If we have students that we know are all over the place, we know that. How do we teach real math when they're all over the map?

Pam Harris  08:42

And that's the crux of teaching real math. 

Kim Montague  08:45

Yeah. 

Pam Harris  08:46

The crux of teaching real math is that if we do, if we teach real mathematics, mathematical reasoning, it opens mathematics for everyone, wherever they are. So the short answer is, we need open tasks, tasks that everybody can enter, everybody has access to, and everybody can learn from. So if I were to describe that with my fingers, remember before I had sort of one finger, and then I had one finger or one finger representing all the students, and one finger representing the tasks, so I've got this kids are all the same place. So I do this one task and everybody mastered that, and that doesn't work. So since that doesn't work, then I sort of splay my fingers open. I'm like kids are everywhere. They're all over here. And so what we need is to be able to have tasks that are also open and so now my other hand, the hand that has been representing the tasks, it also has my finger splayed open. So I've got like fingers going into like multiple fingers. So that if I've got kids kind of all over the place, I do a task that also is open enough for every, all of them to enter, to have access to so that they could all have a place to start, to get off, something that they can make sense of in that problem.

Kim Montague  10:03

So when you say open tasks it feels like a lot. Are you suggesting multiple tasks all the time for all different kids?

Pam Harris  10:13

And because we see that, right? Like, we've gone into some classrooms where teachers have completely individualized programs where they have, like, every kid is doing something completely different. They go to the bucket, and they pull out whatever. And it's all over and the teachers are like trying to, like meet all their needs, write new tasks. And it's -no. 

Kim Montague  10:28

No. 

Pam Harris  10:30

No, the short answer is no, we're not suggesting that. We're actually suggesting that a task can be open enough that all students have some purchase; that all students have something to think about, some way of entering it. And even if they don't - like in a particular Problem String - once the first question in a Problem String is answered, and we've sort of established the answer, we've really thought about it and made the thinking visible. Now students have access to what's about to happen in the next problem in that task, because we've kind of established that first problem. So it's about having something that all students can enter, they can all start to mess with. And then once they've all kind of grown through starting that task, then we up the ante just a little bit so that they all continue to have access. We come in, and we're like, "Well, then what about this?" And we do the next sort of baby step enough that everybody's still learning and everybody's still growing. 

Kim Montague  11:28

Yeah, and I've heard some people kind of describe these kinds of tasks as low floor, high ceiling?

Pam Harris  11:34

Yeah, so not actually my favorite way of describing it. Kind of, it's okay, it's all right. But it doesn't really have to be low floor, it just has to be low enough. So it doesn't have to be like crazy low. 

Kim Montague  11:46

Right.

Pam Harris  11:46

You get to know your students and sort of where they are and kind of what they have access to. And then need tasks that are just low enough for them. We definitely want high ceiling, because we definitely want it high enough that we're sort of encouraging and challenging all students. But the biggest point is not what we call it for sure, the biggest point is that it's not mastery at one shot, that you get to come back around to it and get more sophisticated with it. It's about taking the long view of the most important things, so that it's not like, "Oh, we better get it today. If you don't have it today, then you're not gonna be able to do what we do tomorrow." Instead, what if what we did was more open, but that we have to come back at it, we have to have another shot at it to get more sophisticated.

Kim Montague  12:29

It sounds a little bit like you're saying that as long as we have tasks that are open enough, everyone will get it. That's what people come to you with all the time, right? They say I did this thing, this task, this Problem String, whatever it is. And I'm air quoting. And they say "my kids didn't get it".

Pam Harris  12:46

And so I will respond back to you, which is difficult by email. People ask me about it by email. I'm like - I wish I could show you and here we are on the podcast. Let me see if I can do my best. When they say to me, "My kids didn't get it." I'll say "Oh, it didn't work? What's your it?" Right? Like, it didn't work? They're like, "Yeah, they didn't get it." I'm like, "Okay, but what's your it? It didn't work." And when I do that, if you can see my fingers, I'm back to instead of having my finger sort of splayed out to that open task where there's multiple entry points, now I'm back to one finger. Oh an 'it', just one finger. They're like, "Well, yeah, but  Pam, you know, the Problem String, the task, whatever, it has an outcome. It has a goal." Sure. But it should have multiple outcomes, multiple goals. Not like huge all over the map, but open enough that it's not an 'it' anymore. Like what is your 'it'? So you might be thinking, "I want to do this thing. I want to try real math. I want to teach real math, Pam. That sounds good." But you worry when kids don't get 'it'. Like, for example, let's say you look at a Problem String. And you say to yourself, "Oh, well, they're supposed to learn this strategy." And when you start doing the Problem String, they're not all doing the strategy. And you're thinking, "Oh, no." Because we're used to this sort of mastery approach. We're used to 'it'. And the kids all need to be doing 'it' before we're done with the idea. We do, you do. But that's not the point here. So if I say to myself, "Ooh, they're not getting 'it'." But I've got my fingers all splayed out to represent the kids everywhere. And I'm like, "But they're not getting 'it'." If you're just trying to get your 'it', that's not going to work. So if all the, if the kids are everywhere, get your fingers all over the place, and I've got an open enough task. So those fingers are all over the place. And check it out. The kids now are able to enter, they have access. Everybody learns, everybody grows. My hand, the finger hand, just met the task. And they all went up, they all grew. They all learned it just went up. Well, then I've got to come at it again. I've got to up the ante. And now they all move up again. So I've just moved up my left hand and my right hand sort of met it. I've upped the ante, we do a new task. Everybody learns, everybody grows. I up the ante with a new task. Everybody learns, everybody grows. Now look where you are. Now you've met your 'it'. Now your 'it' was way down there, but because you've kept everybody incrementally going up, now everybody's actually doing 'that thing', but you had to come at it a few times. Whoa, that might sound a little overwhelming. 

Kim Montague  15:02

Yeah. 

Pam Harris  15:02

But notice that as you sort of kept everybody learning and growing, that also meant that the challenging kids were learning and growing, that they were increasing. So even though you sort of met your 'it', the quicker kids or kids that need to be challenged, they are also being encouraged and challenged. It's not about just one little 'it', it's about an open enough task that everybody was learning and growing. So this can be challenging, right? If we're used to teaching to mastery. In fact, just today, in our journey, membership group, one of the people in our fantastic mentoring group mentioned that not all of our students were doing the target strategy during the string, that was the first time she had done a string. But listen to me, as I say, 'the string' to teach the strategy. Oh, it's not about one experience and they own it. It's about taking the long view, developing mathematical reasoning over your entire course. So Problem Strings are so cool, because it keeps all students challenged at the same time that it continues to allow access to all students who might need a little bit of a lower floor. Because in a Problem String, we're dealing with strategy about, really because in real math, as we're teaching real mathematics, there's strategy involved, then that's one way that we keep those students challenged. We don't have students that are bored. So let's say that we've got this Problem String, and we've got kids that need that open access, because they need a purchase. They need a place to enter. But we've also got students who are like, "Yeah, yeah, whatever." Oh, but those students who are like, "Yeah, yeah, whatever." Those students can be challenged, because they get to talk about strategy. And they get to seek for the most clever strategy. It's not about just getting an answer. We want them to be like, "Well, yeah, you got an answer, but how'd you do it? Oh, that way? Well check out how that way. I mean, could you get even more clever?" We want to nudge them to be more clever. So just because they get an answer, they don't stop. And they keep thinking, "Oh, can I do that even in a more clever way?" Or also, we can encourage those students to generalize.

Kim Montague  17:06

Yeah.

Pam Harris  17:06

What does this strategy look like for all problems? Or even cooler! Kim, I've totally seen you do this, where you walk around and you're like, "Find me a problem, where that's not a good strategy." 

Kim Montague  17:16

Right. 

Pam Harris  17:17

And you just walk away. The kid looks at you like, "Oh." They get that look on their face, my eyes just got really big. And that look on their face like, "Oh, whoa, for what problems is this not a good strategy?" And they're off working on that, while you're helping the other students who just barely have a purchase into what's happening or just barely starting to play with a strategy. So you might have kids sort of all over that map, kids that are barely starting to play with the strategy. Which might look like them, by the way, might look like them doing the algorithm or like leaning on something that they know, doing something less sophisticated, like, just partial products instead of something more interesting than that. But they're doing that because that's what they understand. But then they're also hearing the conversations of other students. They're seeing you model their thinking and making it visible, and then connecting it to the more sophisticated strategies. And that gives them a chance to go, "Oh, well, I could do that. I have access to that." And then you help them think, "Oh, I want my brain to do that next time." All students are doing that at some level. Well, frankly, at the level they're at, right? And that's the point like wherever they are, we're helping them do more and more real math. And those fingers just keep moving up. And we don't sort of lose any kids. Because we're doing stuff that's open enough.

Kim Montague  18:34

Yeah, I'm so glad you mentioned that last piece, because just because kids aren't putting on their paper that target strategy doesn't mean they're not participating in some way, by hearing what's going on. We need to give them lots of experiences, maybe to be willing to try something new. Okay, so to summarize, when you teach with open-enough tasks, everyone can learn from where they are and everyone can grow. Then we need to up the ante with a next open enough task and keep everyone learning and growing; everyone learning real math together from where they are, and everyone can be challenged. 

Pam Harris  19:12

Absolutely, super. I'm glad that we had a chance to talk about everybody learning and everybody growing. Remember y'all to join us on MathStratChat on Facebook, Twitter, or Instagram on Wednesday evenings where we explore problems with the world.

Kim Montague  19:28

Hey, if you find the podcast helpful, please rate it and give us a review. We get the biggest kick out of reading your comments. Just like PAmathteacher said not too long ago, "I am blown away each week of your ideas and I leave ready to tell my sixth grade students what I learned." Thanks so much, we really appreciate that.

Pam Harris  19:46

Nice! Thanks PAmathteacher for that review. We'd appreciate it if you guys also would give us a review, that helps more people find the podcast. So if you're interested to learn more math and you want to help yourself and students develop as mathematicians, then don't miss the Math is Figure-Out-Able Podcast because Math is  Figure-Out-Able!